Rod Equation Research Articles

Placement optimization of actuator and sensor and decentralized adaptive fuzzy vibration control for large space intelligent truss structure

Large space truss structure is widely used in spacecrafts. The vibration of this kind of structure will cause some serious problems. For instance, it will disturb the work of the payloads which are supported on the truss, even worse, it will deactivate the spacecrafts. Therefore, it is highly in need of executing vibration control for large space truss structure. Large space intelligent truss system (LSITS) is not a normal truss structure but a complex truss system consisting of common rods and active rods, and there are at least one actuator and one sensor in each active rod. One of the key points in the vibration control for LSITS is the location assignment of actuators and sensors. The positions of actuators and sensors will directly determine the properties of the control system, such as stability, controllability, observability, etc. In this paper, placement optimization of actuators and sensors (POAS) and decentralized adaptive fuzzy control methods are presented to solve the vibration control problem. The electro-mechanical coupled equations of the active rod are established, and the optimization criterion which does not depend upon control methods is proposed. The optimal positions of actuators and sensors in LSITS are obtained by using genetic algorithm (GA). Furthermore, the decentralized adaptive fuzzy vibration controller is designed to control LSITS. The LSITS dynamic equations with considering those remaining modes are derived. The adaptive fuzzy control scheme is improved via sliding control method. One T-typed truss structure is taken as an example and a demonstration experiment is carried out. The experimental results show that the GA is reliable and valid for placement optimization of actuators and sensors, and the adaptive fuzzy controller can effectively suppress the vibration of LSITS without control spillovers and observation spillovers.

Exact Longitudinal Vibration Characteristics of Rods with Variable Cross-Sections

Longitudinal natural vibration frequencies of rods (or bars) with variable cross-sections are obtained from the exact solutions of differential equation of motion based on transformation method. For the rods having cross-section variations as power of the sinusoidal functions ofax+b, the differential equation is reduced to associated Legendre equation by using the appropriate transformations. Frequency equations of rods with certain cross-section area variations are found from the general solution of this equation for different boundary conditions. The present solutions are benchmarked by the solutions available in the literature for the special case of present cross-sectional variations. Moreover, the effects of cross-sectional area variations of rods on natural characteristics are studied with numerical examples.

Exact longitudinal vibration characteristics of rods with variable cross-sections

Longitudinal natural vibration frequencies of rods (or bars) with variable cross-sections are obtained from the exact solutions of differential equation of motion based on transformation method. For the rods having cross-section variations as power of the sinusoidal functions of ax+b, the differential equation is reduced to associated Legendre equation by using the appropriate transformations. Frequency equations of rods with certain cross-section area variations are found from the general solution of this equation for different boundary conditions. The present solutions are benchmarked by the solutions available in the literature for the special case of present cross-sectional variations. Moreover, the effects of cross-sectional area variations of rods on natural characteristics are studied with numerical examples.

Global attractor of strongly damped nonlinearthermoelastic coupled rod system

In this paper, we prove the existence of the global attractor for strongly damped nonlinear thermoelastic coupled rod equation with the initial boundary value, and make the dimension estimate of the global attractor, according to the operator semigroups theory.

On the theory of porous elastic rods

We consider the direct approach to the theory of rods, in which the thin body is modelled as a deformable curve with a triad of rigidly rotating orthonormal vectors attached to every material point. In this context, we employ the theory of elastic materials with voids to describe the mechanical behavior of porous rods. First, we derive the dynamical nonlinear field equations of the model. Then, in the framework of linear theory, we prove the uniqueness of the solution to the associated boundary-initial-value problem. We identify the relevant field quantities from the theory of directed curves by comparison with the three-dimensional equations of straight porous rods. Finally, for orthotropic and homogeneous rods, we determine the constitutive coefficients in terms of the three-dimensional elasticity constants by solving several problems in the two different approaches.

Blowup phenomena for a new periodic nonlinearly dispersive wave equation

AbstractWe first establish the local well‐posedness for a new periodic nonlinearly dispersive wave equation. We then present a precise blowup scenario and several blowup results of strong solutions to the equation. The obtained blowup results on the equation improve considerably recent results on the Camassa‐Holm equation and the rod equation (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A series of new exact solutions of nonlinear evolution equations

AbstractWith the aid of computer symbolic computation system Maple, the generalized auxiliary equation method is first applied to two nonlinear evolution equations, namely, the nonlinear elastic rod equation and (2 + 1)‐dimensional Boiti‐Leon‐Pempinelli equation. As a results, some new types of exact traveling wave solutions are obtained which include bell and kink profile solitary wave solutions, and triangular periodic wave solutions and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Global Conservative Solutions of the Hyperelastic Rod Equation

Global Conservative Solutions of the Hyperelastic Rod Equation

Size effects in Kirchhoff flexible rods

The Kirchhoff equations for flexible rods are extended to be able to describe size effects. Thus, the extended equations are applicable not only for macroscopic rods but also for ultrathin rods whose thickness is close to the material length. It is found that extensional size effects exist in flexible rods, namely, the external forces exerted on both ends of rods will increase as the rods are getting thinner if an identical deformation is remained.

Non-uniform dependence and well-posedness for the hyperelastic rod equation

It is shown that the solution map for the hyperelastic rod equation is not uniformly continuous on bounded sets of Sobolev spaces with exponent greater than 3/2 in the periodic case and greater than 1 in the non-periodic case. The proof is based on the method of approximate solutions and well-posedness estimates for the solution and its lifespan.

Exact traveling wave solutions and bifurcations in a nonlinear elastic rod equation

The exact parametric representations of the traveling wave solutions for a nonlinear elastic rod equation are considered. By using the method of planar dynamical systems, in different parameter regions, the phase portraits of the corresponding traveling wave system are given. Exact explicit kink wave solutions, periodic wave solutions and some unbounded wave solutions are obtained.

Whole analogy between Daniel Bernoulli solution and direct kinematics solution

In this paper, the relationship between the original Euler-Bernoulli's rod equation and contemporary knowledge is established. The solution which Daniel Bernoulli defined for the simplest conditions is essentially the solution of 'direct kinematics'. For this reason, special attention is devoted to dynamics and kinematics of elastic mechanisms configuration. The Euler-Bernoulli equation and its solution (used in literature for a long time) should be expanded according to the requirements of the mechanisms motion complexity. The elastic deformation is a dynamic value that depends on the total mechanism movements dynamics. Mathematical model of the actuators comprises also elasticity forces.

Blow-up of solutions to a periodic nonlinear dispersive rod equation

In this paper, firstly we find an optimal constant for a convolution problem on the unit circle via the variational method. Then by using the optimal constant, we give a new and improved sufficient condition on the initial data to guarantee the corresponding strong solution blows up in finite time. We also analyze the corresponding ordinary difference equation associate to the convolution problem and give numerical simulation for the optimal constant.

Dynamic Higher-Order Equations for Finite Rods

This work considers longitudinal wave propagation in circular cylindrical rods adopting Bostrom's power series expansion method in the radial coordinate. Equations of motion together with consistent sets of general lateral and end boundary conditions are derived in a systematic fashion up to arbitrary order using a generalized Hamilton's principle. Analytical comparisons are made between the present theory to low order and several classic theories. Numerical examples for eigenfrequencies, displacement and stress distributions are given for a number of finite rod structures. The results are presented for series expansion theories of different order and various classical theories, from which one may conclude that the present method generally models the rod accurately.

The Theory of Thin-Walled Rods of Open Profile as a Result of Asymptotic Splitting in the Problem of Deformation of a Noncircular Cylindrical Shell

A new asymptotic approach to the theory of thin-walled rods of open profile is suggested. For the problem of linear static deformation of a noncircular cylindrical shell we consider solutions, which are slowly varying along the axial coordinate. A small parameter is introduced in the equations of the modern theory of shells. Conditions of compatibility for the shell strain measures are employed. The principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms. We conclude the procedure with the subsequent solution for the field of displacements. The analysis shows that the known equations of thin-walled rods, which were previously obtained with some approximate methods using hypotheses and approximations of displacements, are asymptotically exact. The presented semi-numerical analysis of the shell equations allows us to estimate the accuracy of the obtained solution. The results of the paper constitute a sound basis to the equations of the theory of thin-walled rods and provide trustworthy information concerning the distribution of stresses in the cross-section.

Blowup and blowup rate of solutions to a weakly dissipative periodic rod equation

In this paper, we first establish the local well posedness for a weakly dissipative periodic rod equation. We then present a precise blowup scenario and several blowup results of strong solutions to the equation. We finally give the blowup rate of strong solutions to the equation when blowup occurs. Our blowup results for the weakly dissipative periodic rod equation improve considerably earlier results for the periodic rod equation.

Lie Symmetries, Conservation Laws and Exact Solutions for Two Rod Equations

In this paper, the Lie symmetry analysis are performed for the two rod equations. The infinite number of conservation laws (CLs) for the two equations are derived from the direct method. Furthermore, the all similarity reductions and exact explicit solutions are provided.

Wave Breaking and Persistence Properties for the Dispersive Rod Equation

This paper is concerned with some aspects of blow up of solutions and persistence properties. Firstly, we will try to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Then a particular class of initial data for the periodic case is also considered in this paper. Finally, we investigate the persistence properties of the strong solutions.

Blow-up phenomenon for a periodic rod equation

In this paper, firstly we find the optimal constant for a kind of Sobolev inequality on the unit circle via Fourier series. Then we apply the optimal constant on a nonlinear rod equation to give sufficient conditions on the initial datum, which guarantee finite-time singularity formation for the corresponding solution.

Initial boundary value problems for nonlinear dispersive wave equations

In this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half-line and on a finite interval subject to homogeneous Dirichlet boundary conditions. We first prove local well-posedness of the rod equation and of the b-equation for general initial data. Furthermore, we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span. In the case of finite time singularities we are able to describe the precise blow-up scenario of breaking waves. Our approach is based on sharp extension results for functions on the half-line or on a finite interval and several symmetry preserving properties of the equations under discussion.